Smooth surfaces with non-simply-connected complements

Abstract

We consider two constructions of surfaces in simply-connected $4$–manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author [Geom. Topol. 10 (2006) 27–56]. We also construct, for any group $G$ satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group $G$. In each case, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth $s$–cobordism and hence are topologically equivalent for good groups.